Hierarchy of chaotic maps with an invariant measure and their coupling

Hierarchy of one-parameter families of chaotic maps with an invariant measure have been introduced, where their appropriate coupling has led to the generation of some coupled chaotic maps with an invariant measure. It is shown that these chaotic maps (also the coupled maps) do not undergo any period doubling or period-n-tupling cascade bifurcation to chaos, but they have either single fixed point attractor at certain values of the parameters or they are ergodic in the complementary region. Using the invariant measure or Sinai–Ruelle–Bowen measure the Kolmogrov–Sinai entropy of the chaotic maps (coupled maps) have been calculated analytically, where the numerical simulations support the results.